Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions
on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-Neumark theorem
says that the category of measurable spaces is contravariantly equivalent to the category
of commutative von Neumann algebras). Even though we do not have a complete geometric description
for the noncommutative case, we can still use geometric intuition from the commutative case effectively.

A generalization of this idea is given by Grothendieck's relative point of view, which says that a morphism of rings f: A → B
should be regarded geometrically as a bundle of spaces with the total space Spec B fibered over the base space Spec A
and all notions defined for individual spaces should be generalized to such bundles fiberwise.
For example, for von Neumann algebras we have operator valued weights, relative L^p-spaces etc.,
which generalize the usual notions of weight, noncommutative L^p-space etc.

In noncommutative geometry this point of view is further generalized to bimodules.
A morphism f: A → B can be interpreted as an A-B-bimodule B, with the right action of B given by the multiplication
and the left action of A given by f.
Geometrically, an A-B-bimodule is like a vector bundle over the product of Spec A and Spec B.
If a bimodule comes from a morphism f, then it looks like a trivial line bundle with the support being
equal to the graph of f.
In particular, the identity morphism corresponds to the trivial line bundle over the diagonal.

For the case of commutative von Neumann algebras all of the above can be made fully rigorous
using an appropriate monoidal category of von Neumann algebras.

This bimodule point of view is extremely fruitful in noncommutative geometry (think of Jones' index,
Connes' correspondences etc.)

However, I have never seen bimodules in other branches of geometry (scheme theory,
smooth manifolds, holomorphic manifolds, topology etc.) used to the same extent as they are used
in noncommutative geometry.

Can anybody state some interesting theorems (or theories) involving bimodules in such a setting?
Or just give some references to interesting papers?
Or if the above sentences refer to the empty set, provide an explanation of this fact?

8 Answers
8

In "commutative geometry," I think bimodules tend to be a little concealed. People are more likely to talk about "correspondences" which are the space version of bimodules: A correspondence between spaces X and Y is a space Z with maps to X and Y.

When you think in this langauge, there are lots of examples you're missing. For example, the right notion of a morphism between two symplectic manifolds is a Lagrangian subvariety of their product, or even a manifold mapping to their product with Lagrangian image (maybe not embedded). See, for example, Wehrheim and Woodward's functoriality for Lagrangian correspondences in Floer homology

Similarly, correspondences are incredibly important in geometric representation theory. See, for example, the work of Nakajima on quiver varieties.

The theory of stacks also is at least partially founded on taking correspondences seriously as objects, and in particular being able to quotients by any (flat) correspondence.

This same philosophy also underlies groupoidification as studied by the Baez school (they tend to use the word "span" instead of "correspondence" but it's the same thing).

I think groupoidification has more to do with 2-categories of spans, which the 2-category of rings and bimodules is not as far as I know. Still, the ideas are similar as you say.
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Reid BartonOct 24 '09 at 16:26

Well, if you ask me groupoidification is just a hands-on language looking at the finite field (including F_1) points of correspondences of stacks, which makes it look a lot more bimodular.
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Ben Webster♦Oct 24 '09 at 17:00

The Fourier-Mukai transform comes from a bimodule: the Poincaré bundle. Let A be an abelian variety, the Poincaré bundle P is a vector bundle on AxÂ coming from the fact that the points in the dual abelian variety Â parametrize line bundles on A (P is the universal family). In the Fourier-Mukai construction, P is used as a OA-OÂ-bimodule to produce a functor between the derived categories of coherent sheaves on A and Â via a push-pull construction.

Here's a theorem from derived algebraic geometry: if A and B are A∞ algebras (think associative algebras) then giving an A-B-bimodule is the same as giving a functor from {right A-modules} to {right B-modules} which preserves colimits (equivalently, has a right adjoint). The correspondence sends AMB to the functor – ⊗AAMB. Under this correspondence, tensor product of bimodules over the middle algebra is realized by composition of functors.

Most of the theories have some version of Eilenberg-Watts theorem (all (homotopy) cocontinuous functors come from bimodules). They usually also have some version of Mitchell's theorem that characterizes categories of modules. Combined together they give Morita equivalence theory that establishes an equivalence between the bicategory of rings, modules, and intertwiners and the appropriate bicategory of categories, functors, and natural transofrmations. But this is a universal phenomenon. It would be more interesting to find concrete applications in specific cases (like Ben Webster's examples).
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Dmitri PavlovOct 22 '09 at 21:18

1

Can you give a statement of Mitchell's theorem in the DAG case (or a reference)?
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Reid BartonOct 22 '09 at 22:34

I have a feeling that I might have seen something like this in one of Lurie's papers (DAG-I?), but I might be wrong on this issue. An abelian category is enriched over abelian groups, which correspond to spectra in the derived case, so our category should be enriched over spectra (looks like stable (∞,1)-category?). It should also be (homotopy) cocomplete and have some sort of (homotopy) generator. I guess one should look at the proof of the usual algebraic theorem to guess exact conditions that guarantee the existence of an equivalence.
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Dmitri PavlovOct 26 '09 at 17:58

Thanks, this must be Theorem 4.4.9 of DAG II. (It's not quite what I was hoping for, since it refers to objects of the category. I would like a characterization which only refers to the category viewed as an object of the (∞,2)-category of stable presentable (∞,1)-categories. But maybe that's too much to ask.)
–
Reid BartonOct 26 '09 at 18:51

In particular, if the scheme X is semiseparated(say algebraic varieties), M[f] is a
A[U] tensor A[U]^op module(it is A[U]-bimodule]. In other words, G[f] is a coalgebra in the monoidal category of A[U] tensor A[U]^op -modules(A[U]-bimodule)

Reference of Beck's theorem for noncommutative scheme(I mentioned above) is

Maxim Kontsevich and Alexander.L.Rosenberg
Noncommutative spaces and flat descent.(This paper is in Max Plank preprint series, it is online)

3.If one need to define differential operator in general noncommutative space(such as abelian category), in particular, affine scheme. Noncommutative D-module(in particular, quantum D-module), he needs the differential bimodule.
The reference is:
V.A.Lunts and A.L.Rosenberg
Differential Calculus in Noncommutative Algebraic geometry I and II(These are also in Max-Plank preprint series)

In a paper from 1985, Raeburn and J. Taylor describe how to view all elements of H^2(X,Gm) (etale cohomology) as coming from non-unital Azumaya algebras. The construction relies on bimodule theory for these algebras.

Let S be a scheme of positive characteristic p and X an S-scheme. If F denotes the absolute Frobenius, then we can pull back O_X via F. In the affine case, say S=spec k, X=Spec R, this corresponds to tensoring R over F with R, hence we get a bimodule: for r,f\in R we get fr=r^pf. This bimodule is the beginning of the theory of "F-unit crystals" and a positive characteristic version of the Riemann-Hilbert correspondence!
See for example this survey by Emerson-Kisin.

Well, in commutative algebra, you have the fact that any left module is also a right module, so the notion of bimodule can be considered a bit redundant. Any time that an algebraic geometer (or other) uses a module at all, it's generally an A-A bimodule, but we don't think of that, because it's not different form a left A-module or a right A-module, like it is in the noncommutative case.

Well, this is also the case for noncommutative von Neumann algebras: A left M-module is the same thing as a right M^op-module. In particular, an M-N-bimodule is the same thing as a left M⊗N^op-module for an appropriate monoidal structure on von Neumann algebras. This does not imply that we cannot have an interesting theory of bimodules over von Neumann algebras, in fact we do have such a theory.
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Dmitri PavlovOct 22 '09 at 20:17

I believe the point Charles is making is the fact that in commutative land giving some abelian group M the structure of a left A-module, is equivalent to giving it the structure of a right A-module. So there is nothing really new happening there. However in noncommutative land giving M an A-A-bimodule structure is harder since you have to find two different module structures that interact nicely. But this is obviously just a partial answer since it only talks about A-A-bimodules and not A-B-bimodules in general.
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Grétar AmazeenOct 22 '09 at 21:48

Grétar, I do not quite follow your reasoning. In the commutative case we can also have A-A-bimodules where the left and right actions are different and so we get a nontrivial A-A-bimodule.
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Dmitri PavlovOct 26 '09 at 17:50

Thats true. It was written in haste. I guess I meant that they seem to pop up more often, and perhaps more naturally, when dealing with noncommutative things. But as for the reason for that I do not know.
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Grétar AmazeenOct 26 '09 at 19:24